"Considering how depressed she has been since her breakup, it's no surprise that she has finally decided to let him in," another J.Lo insider tattles to Us. "She needs a boost, and he really wants her, so for the first time since the split, she is allowing herself to go with the flow."

To a point, at least. Both mags quote insiders as saying she's worried that they don't have much common ground, from his lack of Latin blood to her parental obligations to 3-year-old twins Max and Emme.

And even though Cooper is said to be "very supportive of the fact that she is a mom," not everyone is on board with his pursuit.

"He has dated a lot of famous women," points out a mole to Us, "and Jennifer's friends are questioning his intentions."

Bradley called time on his two-year romance with Renée Zellweger back in March, and since then has been linked to the likes of Jennifer Aniston and Olivia Wilde.

J.Lo, 42, split with hubby Marc Anthony in July, a decision she supposedly questioned in the aftermath.

Now, however, "She seems very happy and has snapped out of her post-separation gloom," a source explains to People. "She is all about the future now."

Concludes a snitch to Us, "I don't think her relationship with Bradley will become anything long term, but he could be Mr. Right now."